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If \(f^a(x)\) means the function is nested \(a\) times (for example, \(f^2(x)=f(f(x))\) ), what is the smallest value of \(a\) greater than 1 that satisfies\(f(2)=f^a(2)\)?

 

\(f(x) = \left\{ \begin{array}{cl} \frac{x}{21} & \text{ if }x\text{ is a multiple of 3 and 7}, \\ 3x & \text{ if }x\text{ is only a multiple of 7}, \\ 7x & \text{ if }x\text{ is only a multiple of 3}, \\ x+3 & \text{ if }x\text{ is not a multiple of 3 or 7}. \end{array} \right.\)

 Mar 15, 2022
 #1
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f(2)  =  5

f2(2)  = f(5)  =  8

f3(2)  =  f(8)  =  11

f4(2)  =  f(11)  =  14

f5(2)  =  f(14)  =  42

f6(2)  =  f(42)  =  2

f7(2)  =  5

 Mar 16, 2022

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